qiskit.circuit.library.pauli_feature_map

qiskit.circuit.library.pauli_feature_map(feature_dimension, reps=2, entanglement='full', alpha=2.0, paulis=None, data_map_func=None, parameter_prefix='x', insert_barriers=False, name='PauliFeatureMap')

The Pauli expansion circuit.

The Pauli expansion circuit is a data encoding circuit that transforms input data $\vec{x} \in \mathbb{R}^n$, where $n$ is the feature_dimension, as

Here, $S$ is a set of qubit indices that describes the connections in the feature map, $\mathcal{I}$ is a set containing all these index sets, and $P_i \in \{I, X, Y, Z\}$. Per default the data-mapping $\phi_S$ is

The possible connections can be set using the entanglement and paulis arguments. For example, for single-qubit $Z$ rotations and two-qubit $YY$ interactions between all qubit pairs, we can set:

circuit = pauli_feature_map(..., paulis=["Z", "YY"], entanglement="full")

which will produce blocks of the form

┌───┐┌─────────────┐┌──────────┐                                            ┌───────────┐
┤ H ├┤ P(2.0*x[0]) ├┤ RX(pi/2) ├──■──────────────────────────────────────■──┤ RX(-pi/2) ├
├───┤├─────────────┤├──────────┤┌─┴─┐┌────────────────────────────────┐┌─┴─┐├───────────┤
┤ H ├┤ P(2.0*x[1]) ├┤ RX(pi/2) ├┤ X ├┤ P(2.0*(pi - x[0])*(pi - x[1])) ├┤ X ├┤ RX(-pi/2) ├
└───┘└─────────────┘└──────────┘└───┘└────────────────────────────────┘└───┘└───────────┘

The circuit contains reps repetitions of this transformation.

Please refer to z_feature_map() for the case of single-qubit Pauli-$Z$ rotations and to zz_feature_map() for the single- and two-qubit Pauli-$Z$ rotations.

Parameters

• feature_dimension (int) – Number of qubits in the circuit.
• reps (int) – The number of times the evolution layers are repeated.
• entanglement (str |Mapping[int, Sequence[Sequence[int]]] | Callable[[int], str |Mapping[int, Sequence[Sequence[int]]]]) – Specifies the entanglement structure. Can be a string ('full', 'linear', 'reverse_linear', 'circular' or 'sca') or can be a dictionary where the keys represent the number of qubits and the values are list of integer-pairs specifying the indices of qubits that are entangled with one another, for example: {1: [(0,), (2,)], 2: [(0,1), (2,0)]} or can be a Callable[[int], Union[str | Dict[...]]] to return an entanglement specific for a repetition.
• alpha (float) – The Pauli rotation factor, multiplicative to the pauli rotations.
• paulis (list[str] | None) – A list of strings for to-be-used paulis. If None are provided, ['Z', 'ZZ'] will be used.
• data_map_func (Callable[[Parameter], ParameterExpression] | None) – A mapping function for the data x which can be supplied to override the default mapping.
• parameter_prefix (str) – The prefix used if default parameters are generated.
• insert_barriers (bool) – If True, barriers are inserted in between the evolution instructions and Hadamard layers.
• name (str) – The name of the circuit.

Returns

A quantum circuit implementing the Pauli feature map.

Return type

QuantumCircuit

Examples

>>> prep = pauli_feature_map(2, reps=1, paulis=["ZZ"])
>>> print(prep)
     ┌───┐
q_0: ┤ H ├──■──────────────────────────────────────■──
     ├───┤┌─┴─┐┌────────────────────────────────┐┌─┴─┐
q_1: ┤ H ├┤ X ├┤ P(2.0*(pi - x[0])*(pi - x[1])) ├┤ X ├
     └───┘└───┘└────────────────────────────────┘└───┘

>>> prep = pauli_feature_map(2, reps=1, paulis=["Z", "XX"])
>>> print(prep)
     ┌───┐┌─────────────┐┌───┐                                            ┌───┐
q_0: ┤ H ├┤ P(2.0*x[0]) ├┤ H ├──■──────────────────────────────────────■──┤ H ├
     ├───┤├─────────────┤├───┤┌─┴─┐┌────────────────────────────────┐┌─┴─┐├───┤
q_1: ┤ H ├┤ P(2.0*x[1]) ├┤ H ├┤ X ├┤ P(2.0*(pi - x[0])*(pi - x[1])) ├┤ X ├┤ H ├
     └───┘└─────────────┘└───┘└───┘└────────────────────────────────┘└───┘└───┘

>>> prep = pauli_feature_map(2, reps=1, paulis=["ZY"])
>>> print(prep)
     ┌───┐┌──────────┐                                            ┌───────────┐
q_0: ┤ H ├┤ RX(pi/2) ├──■──────────────────────────────────────■──┤ RX(-pi/2) ├
     ├───┤└──────────┘┌─┴─┐┌────────────────────────────────┐┌─┴─┐└───────────┘
q_1: ┤ H ├────────────┤ X ├┤ P(2.0*(pi - x[0])*(pi - x[1])) ├┤ X ├─────────────
     └───┘            └───┘└────────────────────────────────┘└───┘

>>> from qiskit.circuit.library import efficient_su2
>>> prep = pauli_feature_map(3, reps=3, paulis=["Z", "YY", "ZXZ"])
>>> wavefunction = efficient_su2(3)
>>> classifier = prep.compose(wavefunction)
>>> classifier.num_parameters
27
>>> classifier.count_ops()
OrderedDict([('cx', 39), ('rx', 36), ('u1', 21), ('h', 15), ('ry', 12), ('rz', 12)])

References:

[1] Havlicek et al. Supervised learning with quantum enhanced feature spaces, Nature 567, 209-212 (2019).